we-solve (beta)

Econ-605 · economics · Economics

Solving the learning crisis at scale

LLM status Open unverified

Statement

Original atlas ID: D6 (entry 36). Classification: Editorial implementation, coverage, and learning-policy problem.

Original atlas question: Which pedagogy, incentives and technologies produce durable learning when implemented by normal school systems?

Definitions and assumptions

Fix a school system, baseline pupils, a horizon $H$, and a common outcome scale. Policy $\pi=(p,a,s)$ specifies pedagogy or technology $p$, implementation architecture $a$ (staffing, training, supervision, and incentives), and coverage $s$. Treatment versions include time on task and language. Let $Y_{it}(\pi)$ be learning measured by a prespecified assessment with invariant interpretation across treatment groups, and let $C(\pi)$ be discounted real system cost. Follow baseline pupils across schools and nonattendance. Permit peer, teacher-labor-market, and administrative spillovers; state an exposure model or a system-level equilibrium. Assignment must occur at a level that identifies the intended policy comparison.

Formal research question

For feasible pairs $\pi,\pi_0$, identify or bound \[ \tau_t(\pi,\pi_0)=\mathbb E[Y_{it}(\pi)-Y_{it}(\pi_0)], \qquad N_{\lambda}(\pi)=\tau_H(\pi,\pi_0) -\lambda[C(\pi)-C(\pi_0)], \] where $\lambda\geq0$ is an explicitly chosen conversion between learning units and cost per baseline pupil, using the same population normalization for both. Determine which architectures achieve positive, persistent learning effects under ordinary system staffing and budgets. In factorial designs with supported treatment combinations, identify pedagogy-architecture interactions and how they change with coverage; otherwise bound unsupported national-coverage effects using stated model restrictions.

Interpretation and scope

A successful classroom intervention with exceptional personnel does not define the same treatment as a system rollout. Architecture is part of the policy, not nuisance noise. Incentives can change teacher effort, assignment, test preparation, and reporting, so independent measurement matters. Treatment-dependent attendance or testing creates selection; follow-up or missing-outcome bounds are required. Standardized test scores can lose comparability if tests or reference distributions change, and short-term test improvement need not imply durable learning. Cost-effectiveness comparisons must use common outcome and cost units and include implementation demands. The criterion above is a chosen policy objective, not an assumption-free welfare ranking. General-equilibrium effects on teacher recruitment, school choice, and household inputs require separate identification or structural modeling.

Source audit and status

[S1] verifies Indian education experiments and [S2] a 2020 school-management working paper. The intended identity of "Mbiti et al. (2019)" is unresolved: both the six-author input-incentive paper [S3] and the teacher-pay working paper [S4] are verified candidates. Added [S5] is a 2025 scale-adaptation experiment. Thus scaling is already directly studied; the editorial question concerns supported architecture and coverage contrasts, not a universally unexamined intervention class.

References

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  1. Abhijit V. Banerjee, Shawn Cole, Esther Duflo, and Leigh Linden (2007). Remedying Education: Evidence from Two Randomized Experiments in India. The Quarterly Journal of Economics 122(3), 1235--1264. https://doi.org/10.1162/qjec.122.3.1235. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Randomized remedial and computer-assisted education evidence.
  2. Karthik Muralidharan and Abhijeet Singh (2020). Improving Public Sector Management at Scale? Experimental Evidence on School Governance India. NBER Working Paper 28129. https://www.nber.org/papers/w28129. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Large-scale governance evaluation; verified as a working paper.
  3. Isaac Mbiti, Karthik Muralidharan, Mauricio Romero, Youdi Schipper, Constantine Manda, and Rakesh Rajani (2019). Inputs, Incentives, and Complementarities in Education: Experimental Evidence from Tanzania. The Quarterly Journal of Economics 134(3), 1627--1673. https://academic.oup.com/qje/article/134/3/1627/5479257. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Verified thematic candidate on input-incentive complementarities. Abbreviated atlas citation has no title or link and does not uniquely distinguish this paper from the second 2019 Mbiti-led paper supplied below; existence is verified but intended identity remains unresolved.
  4. Isaac Mbiti, Mauricio Romero, and Youdi Schipper (2019). Designing Effective Teacher Performance Pay Programs: Experimental Evidence from Tanzania. NBER Working Paper 25903. https://www.nber.org/papers/w25903. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: Second verified candidate for the ambiguous shorthand; retained as a working-paper version.
  5. Karthik Muralidharan and Abhijeet Singh (2025). Adapting for Scale: Experimental Evidence on Technology-Aided Instruction in India. NBER Working Paper 34205. https://www.nber.org/papers/w34205. Locator: Primary publisher, working-paper, or author record: bibliographic information and abstract; full-text theorem verification is not claimed. Role: More recent experiment explicitly addresses scalability of instruction.

Integrated refinement: ECON-044

Scaling computer-assisted learning with existing school staff

This refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Ordinary staff versus externally supplied implementers. For coverage $s$ and staffing $e\in\{0,1\}$, where $e=1$ is regular school staff, define $Y_i(s,e)$ and full delivery cost $C(s,e)$. Identify \[ \tau^{\rm staff}=\mathbb E[Y_i(s_1,1)-Y_i(s_0,1)],\qquad \Gamma=\tau^{\rm staff} -\mathbb E[Y_i(s_1,0)-Y_i(s_0,0)]. \] Use the same baseline pupils, assessment and horizon. Coverage may change supervision, teacher time and quality in treated and untreated schools; specify these exposures and assignment support. Include training, monitoring, staff opportunity costs and maintenance. A pilot using exceptional personnel need not identify the regular system's scale-up effect or implementation interaction.

Supplied source audit for ECON-044

Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-044, independently of the original entry's reference numbering. [S1]'s evidence-gap chapter directly discusses scaling computer-assisted learning through existing teachers; [S2] corroborates the topic. The saturation/staff contrast is an editorial specification.

References for integrated refinement ECON-044

[S1] Abhijeet Singh, Laia Navarro-Sola, and Philip Oreopoulos (2025). Education Technology. VoxDevLit 20(1), December 2025.

Verified locator / source role: Conclusion/evidence-gap chapter at the linked primary review URL, inspected for the agenda. See the audit conclusion for distinctions between direct agenda support and editorial extensions.

https://voxdev.org/voxdevlit/education-technology/edtech-evidence-gaps

[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.

Verified locator / source role: Topic heading: Education Technology. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.

https://voxdev.org/topic/evidence-gaps-key-unanswered-questions-across-16-topics-development-economics

Integrated refinement: ECON-045

Separating instructional practices from bundled education technology

This refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Separating technology from instructional practice. For supported factorial assignments of technology $T$ and practice $P$, identify $\tau_T(p)=\mathbb E[Y_i(1,p)-Y_i(0,p)]$, $\tau_P(t)=\mathbb E[Y_i(t,1)-Y_i(t,0)]$, and $\iota=\tau_T(1)-\tau_T(0)$. Specify hardware, software, feedback practice, teacher preparation, baseline population and common delayed assessments. Use consistent assignment or a justified exposure design for spillovers and account for missing pupils. Bundle effects alone do not identify components; label unsupported combinations as unidentified rather than infer them from a single technology rollout. Assess persistence and comparable full resource costs rather than immediate assisted scores alone.

Supplied source audit for ECON-045

Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-045, independently of the original entry's reference numbering. [S1] discusses promising instructional technologies and delivery practices, with [S2] as corroboration. This technology--pedagogy factorial decomposition is an editorial extension, not a named open conjecture in the review.

References for integrated refinement ECON-045

[S1] Abhijeet Singh, Laia Navarro-Sola, and Philip Oreopoulos (2025). Education Technology. VoxDevLit 20(1), December 2025.

Verified locator / source role: Conclusion/evidence-gap chapter at the linked primary review URL, inspected for the agenda. See the audit conclusion for distinctions between direct agenda support and editorial extensions.

https://voxdev.org/voxdevlit/education-technology/edtech-evidence-gaps

[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.

Verified locator / source role: Topic heading: Education Technology. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.

https://voxdev.org/topic/evidence-gaps-key-unanswered-questions-across-16-topics-development-economics

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-605-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-605

Supervision constraints, instructional complementarities, and learning at scale

Problem ID: I25-601. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and policy units

The catalogue asks which combinations of pedagogy, implementation architecture, and coverage produce durable learning under ordinary school staffing. Existing experiments already investigate scale and input--incentive interactions [MS,MM]; the question is not whether scale has ever been studied. We solve a model in which finite supervision makes implementation quality endogenous, and separate its predictions from what limited-coverage experiments identify.

The baseline population has mass one and is followed regardless of school attendance. Outcomes use a fixed independent assessment. Coverage is $s\in[0,1]$, supervisory capacity is $A\in[0,\bar A]$, $\bar A>0$, and pedagogy has productivity $\theta\in(0,\kappa]$. An ordinary teacher assigned to the program chooses implementation effort $e\in[0,1]$ to maximize \[ \theta e-\frac12\left(\kappa+\frac{\delta s}{A}\right)e^2, \qquad\kappa,\delta>0, \] when $s,A>0$. The first term represents professional or incentive value; the second is a declared effort friction, increasing when supervision is spread thin. Set implementation to zero if $A=0$. This is a behavioral assumption, not a claim that teacher preferences can be recovered from test scores.

A fraction $r_H\in(0,1]$ of the initial skill gain remains at the chosen delayed horizon. System-wide instructional displacement costs $\gamma s^2/2$ learning units, with $\gamma>0$. Real delivery cost per baseline pupil is $C(s,A)=cs+kA^2/2$, where $c,k>0$; $c$ includes regular staff time, equipment, maintenance, and their opportunity costs, while the quadratic term represents expanding supervisory capacity. Private effort disutility is not added to this educational cost-effectiveness objective. Fix a conversion $\lambda>0$ from resource costs into learning units. The target is delayed learning minus $\lambda C$, rather than an unqualified social-welfare measure.

Implementation and the optimal scale--architecture pair

Put $B=r_H\theta^2$. Teacher optimization gives \[ e(s,A)=\frac{\theta A}{\kappa A+\delta s},\qquad G(s,A)=\frac{BsA}{\kappa A+\delta s}-\frac\gamma2s^2, \] \[ N(s,A)=G(s,A)-\lambda cs-\frac{\lambda k}{2}A^2. \tag{1} \] Define the fraction in (1) to be zero at $(0,0)$ and on either axis. It is continuous on the whole feasible rectangle.
Theorem (A complete optimality characterization): The objective (1) has a unique maximizer. It is $(0,0)$ if and only if $B/\kappa\le\lambda c$. Otherwise both its coordinates are positive. At a positive maximizer, writing $D=\kappa A+\delta s$, the conditions are \[ \frac{B\kappa A^2}{D^2}-\gamma s-\lambda c \begin{cases}=0,&s<1,\\\ge0,&s=1,\end{cases} \] \[ \frac{B\delta s^2}{D^2}-\lambda kA \begin{cases}=0,&A<\bar A,\\\ge0,&A=\bar A.\end{cases} \tag{2} \] These conditions are sufficient as well as necessary. For every fixed $s>0$, the optimal capacity is the smaller of $\bar A$ and the unique positive root of \[ \lambda kA(\kappa A+\delta s)^2=B\delta s^2. \tag{3} \]
Proof: The teacher objective is strictly concave. Its unconstrained maximizer is the displayed $e$, which belongs to $[0,1]$ because $\theta\le\kappa$. Multiplying true instructional productivity $\theta e$ by coverage and retention gives $G$ after the declared displacement term.

For $s,A>0$, the Hessian of $f(s,A)=sA/(\kappa A+\delta s)$ is \[ -\frac{2\kappa\delta}{(\kappa A+\delta s)^3} \begin{pmatrix}A^2&-sA\\-sA&s^2\end{pmatrix}. \] Its quadratic form is a nonpositive multiple of $(Ax-sy)^2$. Hence $f$ is concave in the positive quadrant and, by continuity and approximation from within that quadrant, on its closure. Subtracting the strictly positive quadratic penalties in both coordinates makes $N$ strictly concave. Continuity and compactness give existence, and strict concavity gives uniqueness.

Since $f\le s/\kappa$, if $B/\kappa\le\lambda c$, (1) is nonpositive and is zero only at the origin. Conversely, if $B/\kappa>\lambda c$, choose a finite $t>0$ so $Bt/(\kappa t+\delta)>\lambda c$. Along $A=ts$ for small positive $s$, the coefficient of $s$ in $N$ is positive, while the remaining costs are quadratic. Such a point is feasible and has positive value. Points on either axis other than the origin have negative value, so the maximizing coordinates are both positive.

Differentiation gives (2). Their signs are exactly the feasible one-sided first-order conditions at the upper endpoints. Concavity makes them sufficient: the tangent-plane inequality bounds every feasible alternative by its value at the candidate. Finally, for fixed $s$, the derivative in $A$ is positive at zero and strictly decreasing, and its zero is (3). The left side of (3) increases continuously from zero to infinity, proving existence and uniqueness of the root and the clipping rule. ∎

This theorem makes staffing and supervision part of the chosen intervention. It does not assume that an effort level observed under exceptional implementation can be supplied at arbitrary coverage. Equation (3) is a one-dimensional monotone root problem, while the joint problem is a strictly concave program.

How naive extrapolation fails

Proposition (An exact pilot-to-scale error): Hold capacity $A>0$ fixed and compare a pilot coverage $s_p>0$ with $S>s_p$. Extrapolating its average gain linearly gives an upward error \[ \frac{S}{s_p}G(s_p,A)-G(S,A) =\frac{BSA\delta(S-s_p)}{(\kappa A+\delta s_p)(\kappa A+\delta S)} +\frac\gamma2S(S-s_p)>0. \tag{4} \] If exceptional external implementers instead maintain effort $\theta/\kappa$ at every coverage, their gross instructional gain is $Bs/\kappa$. The ordinary-staff gross gain has a strictly smaller increase between any two positive coverage levels.
Proof: Subtract the two expressions in (1), before resource costs, and combine denominators to obtain (4). Ordinary gross gain has derivative $B\kappa A^2/(\kappa A+\delta s)^2$, strictly below $B/\kappa$ for $s>0$. Integrating over the coverage interval proves the staffing comparison. The displacement term can be held common when comparing the two implementations and then cancels from their difference in changes. ∎
Proposition (Pedagogy and architecture complement each other): At a fixed $s>0$, let $0<\theta_0<\theta_1\le\kappa$ and $0\le A_0<A_1\le\bar A$, with all other primitives common. The factorial interaction in delayed learning is strictly positive and equals \[ r_H(\theta_1^2-\theta_0^2)s \left[\frac{A_1}{\kappa A_1+\delta s} -\frac{A_0}{\kappa A_0+\delta s}\right]. \] If delivery costs are additive in the pedagogy and capacity components, the same interaction holds for the cost-adjusted criterion.
Proof: Take the difference of pedagogy contrasts at the two capacities. The displacement term cancels. The bracket is positive because $A\mapsto A/(\kappa A+\delta s)$ has derivative $\delta s/(\kappa A+\delta s)^2>0$. An additive cost has zero factorial interaction, proving the final statement. ∎
A bundled technology-versus-control experiment does not identify this interaction: all four supported combinations are needed unless one imposes the structural formula. Independent assessment matters because effort directed at the program's assisted tasks need not equal unassisted retained learning.

What coverage experiments identify

Under the model, fixed known $A$ and $\gamma$ imply that the adjusted per-participant gain $\ell(s)=[G(s,A)+\gamma s^2/2]/s$ obeys \[ \frac1{\ell(s)}=\frac\kappa B+\frac\delta{BA}s. \tag{5} \] Thus exact gains at two distinct positive coverages identify the intercept and slope in (5), and hence predict the same-architecture coverage curve. They do not separately identify $B,\kappa,\delta$: multiplying all three by a common positive factor preserves those observations. Equation (5) is a testable restriction once additional coverage points are observed, rather than a consequence of random assignment itself.

There is also a weaker shape-based conclusion. Suppose an experiment identifies a learning response $g$ up to coverage $s_p$, with $g$ nondecreasing and concave, and identifies its finite left derivative $d\ge0$ at $s_p$. Among all nondecreasing concave continuations, the sharp interval at $S>s_p$ is \[ [g(s_p),\ g(s_p)+d(S-s_p)]. \tag{6} \] Monotonicity gives the lower bound and the concave supporting slope gives the upper bound. Extending after $s_p$ by any constant slope in $[0,d]$ preserves concavity and attains every value in (6). Thus even an exactly measured pilot curve need not point-identify national coverage under shape restrictions alone.

These results assume fixed outcome units, baseline-pupil follow-up, and identified system-level exposure contrasts. Household responses, teacher recruitment, missing assessments, uncertain retention, and national administrative changes require additional evidence. They remain part of the original problem.

Completion Estimate

49%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves an endogenous teacher-effort model with unique optimal coverage and supervision, quantifies pilot extrapolation error, and proves instructional complementarities and sharp shape-based coverage bounds. Durable system learning still requires identified ordinary-staff implementation and retention, supported technology-practice interactions, complete pupil follow-up, costs and teacher or household equilibrium responses.

References

  • [MS] K. Muralidharan and A. Singh (2025), "Adapting for Scale: Experimental Evidence on Technology-Aided Instruction in India," NBER Working Paper 34205. Authors' October 2025 manuscript.
  • [MM] I. Mbiti, K. Muralidharan, M. Romero, Y. Schipper, C. Manda and R. Rajani (2019), "Inputs, Incentives, and Complementarities in Education: Experimental Evidence from Tanzania," Quarterly Journal of Economics 134(3), 1627--1673. doi:10.1093/qje/qjz010.
  • [Ed] A. Singh, L. Navarro-Sola and P. Oreopoulos (2025), "Education Technology," VoxDevLit 20(1), evidence-gap chapter. Primary review. These sources motivate the implementation questions; the equations above are the explicitly declared model.

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